preface: I don't do math. I sometimes look up things in math and physics, and find things of interest. So, in the case of 'squaring the circle', I had a non-mathematical idea about solving this, in part because it has a 'physical' element: "in a finite number of steps, using a compass and straight edge".
I am assuming both tools are marked with increments/there are some ways to determine length along the straight edge, and radius of compass arm separation.
Given the following figure:
Aren't the cardinal directions and the corners of the circle relative to concentric circles within and outside of the first circle?
Or, another way: aren't the intersections of the square and circle plottable along the diameter?